I have connected an external drive to my Airport Extreme, but it doesn't show up in Finder, and it doesn't show up in the list of drives when I connect to the Airport Utility.

The same drive worked previously, but I have since reformatted it. The drive works fine when connected directly to my Mac via USB cable.

Update
I don't recall how the drive was previously formatted, except that I had both a Mac and Windows partition, and the Airport Extreme recognised both of them. Now I've formatted it as a single partition, with format: Mac OS Extended (Case-sensitive, Journaled, Encrypted)

I would probably reformat the drive as journaled HFS+ but not Case Sensitive. Encryption is fine, but Case Sensitive is a pain in the arse, especially because the OS X is generally not case specific.

You plugged the USB drive into the AirPort Extreme, but did you use the AirPort Utility to configure everything to work together?

The rightmost tab/button is where you configure attached storage. You'll need to set the password security for either plain visibility when you useFinder to log into the AirPort Extreme, or if the hard drive will require its own password, independent of the device password.

Whole the setup parameters are fairly straight-forward, I may have missed one or two checkboxes. I'm doing this from memory.

Given the same drive worked previously, I think the format is most likely to be the issue, as covered in other answers here. However, I have come across the "drive doesn't show in AirPort Utility or anywhere else" issue with my AirPort Extreme 802.11n (2nd Generation) (purchased 2007) a few times and thought I'd share what I'd found:

This AirPort Extreme model doesn't provide a lot of power to the USB port. Some drives spun up, and the power light on the drive lit up, but the drive didn't show up in the AirPort Utility. I fixed this issue by providing power to the drive via a split USB cable and mains USB power supply.

For one particular drive, I found the controller board in the drive enclosure was incompatible with the AirPort Extreme. The drive didn't show up in AirPort Utility. When I moved the drive into a different enclosure, it all worked OK.